Theorem upgrfn

Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	isupgr.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
Assertion	upgrfn	$$⊢\left(G\in \mathrm{UPGraph}\wedge EFnA\right)\to E:A⟶\left\{x\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$

Proof

Step	Hyp	Ref	Expression
1		isupgr.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		isupgr.e	$$⊢E=\mathrm{iEdg}\left(G\right)$$
3	1 2	upgrf	$$⊢G\in \mathrm{UPGraph}\to E:\mathrm{dom}E⟶\left\{x\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$
4		fndm	$$⊢EFnA\to \mathrm{dom}E=A$$
5	4	feq2d	$$⊢EFnA\to \left(E:\mathrm{dom}E⟶\left\{x\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}\leftrightarrow E:A⟶\left\{x\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}\right)$$
6	3 5	syl5ibcom	$$⊢G\in \mathrm{UPGraph}\to \left(EFnA\to E:A⟶\left\{x\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}\right)$$
7	6	imp	$$⊢\left(G\in \mathrm{UPGraph}\wedge EFnA\right)\to E:A⟶\left\{x\in \left(𝒫V\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$